A137481 Triangular sequence based on the coefficients of the magnetic model for q=1/2: p(x,t)=Exp[x*t]*((t^2 + 1/2 - 1)/(2*t + 1/2 - 2))^2.

1, 8, 3, 24, 48, 9, -192, 216, 216, 27, -2976, -2304, 1296, 864, 81, -57600, -44640, -17280, 6480, 3240, 243, -1336320, -1036800, -401760, -103680, 29160, 11664, 729, -36126720, -28062720, -10886400, -2812320, -544320, 122472, 40824, 2187, -1114767360, -867041280, -336752640, -87091200

Offset

1,2

Comments

Row sums are: {1, 11, 81, 267, -3039, -109557, -2837007, -78266997, -2424502719, -84168178677, -3241001149551};

These magnetic models are important in the application of nonlinear complex dynamics to physical systems.

References

Peitgen and Richter, eds., The Beauty of Fractals, Springer-Verlag, New York, 1986, page145.

Links

Table of n, a(n) for n=1..40.

Formula

p(x,t)=Exp[x*t]*((t^2 + 1/2 - 1)/ (2*t + 1/2-2))^2=Sum[P(x,n)*t^n/n!,{n,0,Infinity}]; out_n,m=3^(n + 2)*n!*Coefficient(P(x,n))

Example

{{1},
{8, 3},
{24, 48, 9},
{-192, 216, 216, 27},
{-2976, -2304, 1296, 864, 81},
{-57600, -44640, -17280, 6480, 3240, 243},
{-1336320, -1036800, -401760, -103680, 29160, 11664,729},
{-36126720, -28062720, -10886400, -2812320, -544320, 122472, 40824, 2187}, {-1114767360, -867041280, -336752640, -87091200, -16873920, -2612736, 489888, 139968, 6561},
{-38645268480, -30098718720, -11705057280, -3030773760, -587865600, -91119168,-11757312, 1889568, 472392, 19683},
{-1486356480000, -1159358054400, -451480780800, -117050572800, -22730803200, -3527193600, -455595840, -50388480, 7085880, 1574640, 59049}

Mathematica

p[t_] =Exp[x*t]*((t^2 + 1/2 - 1)/(2*t + 1/2 - 2))^2; Table[ ExpandAll[3^(n + 2)*n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[3^(n + 2)*n!* CoefficientList[SeriesCoefficient[ Series[p[t], {t, 0, 30}], n], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Sequence in context: A228313 A037206 A065530 * A228886 A182160 A213838

Adjacent sequences: A137478 A137479 A137480 * A137482 A137483 A137484

Keyword

tabl,sign

Author

Roger L. Bagula, Apr 27 2008

Status

approved